Elasticity of two-dimensional crystalline monolayers of fatty acid salts at an air–water surface

The elastic properties of organic–inorganic two-dimensional crystals floating at the water surface have been fully characterized by grazing incidence X-ray diffuse scattering and high-resolution diffraction. We show that the strong interaction between the organic molecules and the inorganic divalent cations is enough for these nm thick crystals to behave like true solids, with a residual tension of 1 × 10⁻⁴–10⁻³ N m⁻¹. Their bending rigidity is renormalized as κ(q) ∝ q^−η_k with η_k = 0.25 ± 0.07 and a microscopic value ≈ 100 k_BT at q = 1 × 10⁹ m⁻¹. The in-plane elastic constants behave like q^η_u with η_u = 1.41 ± 0.2, obeying the scaling relation η_u = 2 − 2η_k. These results are consistent with a long-range phonon-mediated interaction between out-of-plane fluctuations but the values of the exponents differ from those generally obtained in numerical simulations.